MEMO 2017 ekipno problem 5

Let be an acute-angled triangle with and circumcircle . Let be the midpoint of the shorter arc of , and let be the intersection of the rays and . Let be the intersection of the internal bisector of the angle and the circumcircle of the triangle . Let us assume that is inside the triangle and there is an intersection of the line and the circle such that is the midpoint of the segment .Show that is the midpoint of the segment , where and are the excentres of opposite to and , respectively.

Let $ABC$ be an acute-angled triangle with $AB > AC$ and circumcircle $\Gamma$. Let $M$ be the midpoint of the shorter arc $BC$ of $\Gamma$, and let $D$ be the intersection of the rays $AC$ and $BM$. Let $E \neq C$ be the intersection of the internal bisector of the angle $ACB$ and the circumcircle of the triangle $BDC$. Let us assume that $E$ is inside the triangle $ABC$ and there is an intersection $N$ of the line $DE$ and the circle $\Gamma$ such that $E$ is the midpoint of the segment $DN$.\\
Show that $N$ is the midpoint of the segment $I_B I_C$, where $I_B$ and $I_C$ are the excentres of $ABC$ opposite to $B$ and $C$, respectively.